Question

Four digit numbers are formed using the digits $1,2,3,4$ (repetitions among the digits allowed). Find the number of such four digit numbers divisible by $11$

Answer 1

Let $abcd$ be the number.
If $a=b$ then $c=d$.

There are $4\times 4=16$ ways ---(1)

Which can occur. (Four options for $a$ and four options for $b$.

If $a=b\pm 1$ then $c=d\pm 1$.

And there are $2\times 3\times 3=18$ ways.---(2)

this can occur. (Two choices whether $a>b$ or $b>a$ and three choices from $1,2,3,4$ that are one apart.
If $a=b\pm 2$ then $c=d\pm 2$

and there are $2\times 2\times 2=8$ ways. ---(3)
And if $a=b\pm 4$ then

either $a=1;b=4;c=4;d=1$ (or)

$a=4;b=1;c=1;d=4$.

Which are $2$ ways. ---(4)

From (1), (2), (3) and (4)

Required number of ways $=16+18+8+2=44$ ways.